A kinetic model of the transformation of a micropatterned amorphous precursor into a porous single crystal

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Author: Sofia Magkiriadou, Fall 2011


The authors study the self-assembly of large single crystals with complex shapes. Based on the experimental observation that such materials often come from the crystallization of an amorphous precursor, they provide a simple thermodynamic model for this process which shows how the geometry of the precursor may affect the number of domains in the final product. Many organisms have been observed to form organs, such as teeth or exosceletons, from large single-domain crystals. This process has recently been replicated in the lab using amorphous calcium carbonate to make calcite crystals. Remarkably, it involves the transformation of a substance into a much denser one. At first glance, one would think that the resulting stresses would easily break the crystallization fronts and disrupt the formation of a single domain. Intriguingly, this process has been observed to succeed when the precursor is perforated by holes with a specific range of size and spacing. It is thus believed that the initial geometry plays a crucial role in the successful formation of large single crystals.

Theoretical Model

The authors believe that an important initial condition is that the precursor be a soft material which can deform without maintaining high stresses as the crystal forms. This is indeed the case with amorphous calcium carbonate, which has a lot of water and is relatively flexible. As a model system they use a quasi-two-dimensional layer of

amorphous calcium carbonate, of thickness d, containing holes of

initial radius ρ_wo ad final radius ρ_w on a square lattice with

constant a (see Figure 2).

<Figure 2.>

Initially, it is asserted that in the middle of a square occurs a

small circular crystallization nucleus with radius ρ_c. As the

crystal grows, water is released, which results in the enlargment

of the holes ρ_w. The dynamics of what follows must obey two

conservation laws: calcium ions must be conserved, as must the

number of water molecules. These two facts, in combination with the

universal condition that the change in Gibbs free energy must

decrease during a self-assembly process, leads to a condition on

the maximum hole size, ρ, in the precursor for the formation of a

single crystalline domain. If the holes are smaller than ρ, they

prevent the growth of the crystallization center whereas if they

are larger than ρ their existence does not affect the growth of the

crystallization center. Thus, if the precursor is perforated with

holes smaller than ρ, the number of nucleation sites allowed to

evolve into crystalline domains will be very small and potentially

only one.

In order to gain further understanding into the dynamics of this

process, the authors consider the diffusion of water from the

amorphous calcium carbide away from the material, as the calcium

crytals have no water in them. In particular, they consider the

case where the holes are spaced closely enough that water

preferentially leaves the system through them and not through the

upper and lower surfaces, i.e. the case where a/d is sufficiently


<Figure 3.>

To understand the dynamics the relevant principles are the energy

dissipated during the diffusion of water, comined with Onsager's

principle which requires that this energy dissipation balance the

change in Gibbs free energy of the system (Ref[1]). Combining these

considerations, the authors arrive at a relation between the

diameter of the crystallization nucleus to the time it takes for it

to grow until the next nearest hole. Interestingly, they observe

that this time scales with the square of the lattice size, as

opposed to linearly with it. The implication of this observation to

the kinematics is that, if the holes in the precursor are too

scarce, there is enough time for many crystalline domains to form

before each crystal front reaches the nearest hole, thus preventing

the formation of a single crystal.